Structure Formation in Generalized Rastall Gravity

Recently a modified version of Rastall gravity has been proposed in which a varying coupling parameter could play the role of dark energy (DE) and thus is responsible for the current accelerated expansion of the Universe. Motivated by this modification, we study here the evolution of matter perturbations, in both linear and non-linear regimes, using the top-hat spherical collapse model. The exact solutions in linear regime show that as the Universe evolves, matter perturbations grow and reach a maximum value at a certain redshift after which the perturbations start decreasing at later times. Depending on model parameters, exact oscillatory solutions (in linear regime) can be found representing that matter perturbations could experience either overdense and underdense regions during the dynamical evolution of the Universe. Numerical solutions in non-linear regime show that the amplitude of perturbations grow much faster than the linear one and diverges at a critical redshift. The growth rate of non-linear perturbations and the critical redshift depend crucially on model parameters. It is found that the running mutual interaction between matter and geometry, encoded in the variable Rastall coupling parameter, could affect on the dynamics of matter perturbations during the evolution of the Universe.